Topological dynamical systems induced by polynomials and combinatorial consequences (2301.07873v1)

Abstract: Let $d\in {\mathbb N}$ and $p_i$ be an integral polynomial with $p_i(0)=0$, $1\le i\le d$. It is shown that if $S$ is piecewise syndetic in $\mathbb Z$, then $${(m,n)\in{\mathbb Z}^2: m+p_1(n),\ldots,m+p_d(n)\in S}$$ is piecewise syndetic in ${\mathbb Z}^2$, which extends the result by Glasner and Furstenberg for linear polynomials. Our result is obtained by showing the density of minimal points of a dynamical system of ${\mathbb Z}^2$ action associated with the piecewise syndetic set $S$ and the polynomials ${p_1,\ldots,p_d}$. Moreover, it is proved that if $(X,T)$ is minimal, then for each non-empty open subset $U$ of $X$, there is $x\in U$ with ${n\in {\mathbb Z}: T^{p_1(n)}x\in U, \ldots, T^{p_d(n)}x\in U}$ piecewise syndetic.